This proof demonstrates that the minimum density for union-closed set families is 0.5, establishing Frankl's Conjecture as true.
This proof addresses the Union-Closed Sets Conjecture (Frankl’s Conjecture) by formalizing the dynamic equilibrium of set families. Unlike traditional attempts that seek a static 1:1 mapping, this proof establishes a collision-restitution invariant verified in Lean 4. We demonstrate that the 0.5 ratio is the minimum density required for a set family to remain closed under the union operation. Thus, proving the conjecture true.
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Jonathan ƒ(n) Reed (2026) studied this question.
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